Physics 121 - Electricity and Magnetism Lecture 08 - Multi-Loop and RC Circuits Y&F Chapter 26 Sect. 2 - 5 Definitions of Circuit Terms Kirchhoff Rules Problem solving using Kirchhoff’s Rules Multi-Loop Circuit Examples RC Circuits Charging a Capacitor Discharging a Capacitor Discharging Solution of the RC Circuit Differential Equation The Time Constant Examples Charging Solution of the RC Circuit Differential Equation Features of the Solution

Definitions of Circuit Terms ECE Basic circuit elements have two terminals. 1 2 Node: A point where 2 or more circuit elements are joined. Generic symbol: Essential Node (Junction): A node where at least 3 circuit elements are joined. Path: A route (or trace) through adjacent basic circuit elements with no element included more than once. A path may pass through essential and/or non-essential nodes. It may or may not be closed i i1 i2 Branch: A path that connects two nodes (essential or not). Includes 1 or more elements Essential Branch: A path that connects two essential nodes without passing through another essential node. The circuit elements of an essential branch are all in series. Non-essential (trivial) nodes can connect the circuit elements in “series”. There is exactly one current per essential branch. i Loop: A closed path whose last node is the same as the starting node. Mesh: A loop that does not enclose any other loops. Planar Circuit: A circuit whose diagram can be drawn on a plane with no crossing branches.

Examples of Circuit Terms Source: Nilsson & Reidel Electric Circuits - + v2 i6 v1 R1 R2 R3 R4 R6 R5 R7 b a d c f e g i5 i4 i3 i2 i1 i7 Nodes: a, b, c, d, e, f, g (nodes b & g are drawn extended) Essential Nodes (Junctions): b, c, e, g Branches: v1, v2, R1, R2, R3, R4, R5, R6, R7, i6 Essential Branches (7): v1-R1, R2-R3, v2-R4, R5, R6, R7, i6 Currents: 1 per essential branch, total of 7: i1 . . . i7 Ideal, Independent Voltage Sources (ideal EMFs) : v1, v2 Ideal, Independent Current Source: i6 Meshes (subset of loops): v1-R1-R5-R3-R2, v2-R2-R3-R6-R4, R5-R7-R6, R7-i6 Loops: Meshes + V1-R1-R5-R6-R4-V2, V1-R1-R7-R4-V2, v1-R1-i6-R4-V2 , R6-R5-i6 Paths: Many, traversing 1, 2, …..n branches

Solving Circuit Problems using Kirchhoff’s Rules CIRCUITS CONSIST OF: JUNCTIONS (ESSENTIAL NODES) ... and … ESSENTIAL BRANCHES (elements connected in series, one current/branch) i The current through all series elements in an essential branch is the same; i.e., the number of currents = the number of essential branches Often, currents are the unknowns, all other elements are specified Analysis strategy: 1) Use Kirchhoff Rules to generate N independent equations in N unknowns Loop Rule (energy conservation): The algebraic sum of all potential changes is zero for every closed path around a circuit. Corollary: voltage difference is the same for all paths connecting a pair of nodes. Junction Rule or Current rule or Node Law (charge conservation): At any junction the algebraic sum of the currents at any (essential) node equals zero 2) Solve the resulting set of simultaneous equations (you need a strategy). Linear, algebraic for resistances and EMFs only (now)

Applying Kirchhoff’s Rules: “active” sign convention: voltage drops considered negative “passive” sign convention: voltage drops considered positive “branch” means “essential branch “junction” means “essential node” Procedure for Generating Circuit Equations Name the currents or other unknowns. In each branch, arbitrarily assign direction to current. - A negative result  opposite flow. Apply Junction Rule, create equations Apply Loop create equations: - Choose direction for traversing each closed loop. possibly traverse branches with or against assumed current directions. When crossing resistances: - Voltage drop (DV = - iR) is negative when following assumed current. - Positive voltage change DV = +iR for crossing opposite to assumed current. When crossing EMFs from – to +, DV = +E. Otherwise DV= -E. Keep generating equations until you have N independent ones. After solving, calculate power or other quantities as needed. Dot product i.E determines whether EMFs supply or dissipate power For later: When following current across C write –VC= -Q/C. When crossing inductance write VL= - Ldi/dt

Example: Equivalent resistance for resistors in series Junction Rule: The current through all of the resistances in series (a single branch) is identical. No information from Junction/Current/Node Rule Loop Rule: The sum of the potential differences around a closed loop equals zero. Only one loop path exists: The equivalent circuit replaces the series resistors with a single equivalent resistance: same E, same i as above CONCLUSION: The equivalent resistance for a series combination is the sum of the individual resistances and is always greater than any one of them. inverse of series capacitance rule

Example: Equivalent resistance for resistors in parallel Loop Rule: The potential differences across each of the 4 parallel branches are the same. Four unknown currents. Apply loop rule to 3 paths. i not in these equations Junction Rule: The sum of the currents flowing in equals the sum of the currents flowing out. Combine equations for all the upper junctions at “a” (same at “b”). The equivalent circuit replaces the series resistors with a single equivalent resistance: same E, same i as above. CONCLUSION: The reciprocal of the equivalent resistance for a parallel combination is the sum of the individual reciprocal resistances and is always smaller than any one of them. inverse of parallel capacitance rule

EXAMPLE: MULTIPLE BATTERIES SINGLE LOOP + - R1= 10 W R2= 15 W E1 = 8 V E2 = 3 V i + A battery (EMF) absorbs power (charges up) when I is opposite to E E2 is opposite to Vdrop -3.0x0.2

Example: Multi-loop circuit with 2 EMFs Given all resistances and EMFs in circuit: Find currents (i1, i2, i3), then potential drops and power dissipated by resistors 3 unknowns (currents) imply 3 independent equations needed R1 E1 i1 + - E2 R3 R2 i3 i2 A B C D E F Apply Procedure: Identify essential branches (3) & junctions (2). Name all currents (3) and other variables. Same current flows through all elements in any series branch. Assume arbitrary current directions; negative result means opposite direction. At junctions, write Current Rule (Junction Rule) equations. Same equation at junctions A and B (not independent). Junction Rule yields only 1 of 3 equations needed Are points C, D, E, F junctions? (not essential nodes)

Procedure, continued: + - Solution: (after a lot of algebra) Apply Loop Rule as often as needed to find equations that include all the unknowns (3). Traversal direction is arbitrary. When following the assumed current direction voltage change = - iR. When going against assumed current direction voltage steps up by +IR EMF’s count positive when traversed from – to + side EMF’s count negative when traversed from + to - sides R1 E1 i1 + - E2 R3 R2 i3 i2 A B C D E F ADCBA - CCW ADCBFEA - CCW ABFEA - CCW Only 2 of these three are independent Now have 3 equations in 3 unknowns Loop equations for the example circuit: Solution: (after a lot of algebra) Define:

Example: find currents, voltages, power 6 BRANCHES  6 CURRENTS. JUNCTION RULE: Junctions C & E are the same point, as are D & F -> 4 currents left. Remaining 2 junction equations are dependent -> 1 junction equation ABCDA - CW CEFDC - CW EGHFE - CW LOOP RULE: FINISH: POWER:

Multiple EMF Example: find currents, voltages, power R1 = 2 W R2 = 4 W E1 = 3 V E2 = 6 V MULTIPLE EMF CIRCUIT USE THE SAME RULES JUNCTION RULE at A & B: LOOP ACDBA: LOOP BFEAB: USE JUNCTION EQUATION: EVALUATE NUMERICALLY: For power use:

Can constant current flow through a capacitor indefinitely? RC Circuits: Time dependance 27.8 + - E i C R a b Vc Can constant current flow through a capacitor indefinitely? Capacitance, Resistance, EMF given Use Loop Rule + Junction Rule New Loop Rule term: Vc= Q/C Find Q, i, Vc, U for capacitor as functions of time First charge up C (switch to “a”) then discharge (switch to “b”) Charging: “Step Response” Switch to “a” then watch. Loop equation: Assume clockwise current i through R As t  infinity: Vcap E, Q  Qinf = CE Expect zero current as t  infinity Expect largest current at t = 0, Energy is stored in C, some is dissipated in R Discharging: Switch to “b”. no EMF, Loop equation: Energy stored in C is now dissipated in R Arbitrarily chose current still CW Vcap= E at t =0, but it must die away Q0= full charge = CE Result: i through R is actually CCW

RC Circuit: solution for discharging Circuit Equation: Loop Equation is : Substitute : First order differential equation, form is Q’ = -kQ  Exponential solution Charge decays exponentially: t 2t 3t Q0 Q t/RC is dimensionless RC = t = the TIME CONSTANT Q falls to 1/e of original value Voltage across C also decays exponentially: Current also decays exponentially:

Time t 2t 3t 4t 5t Value e-1 e-2 e-3 e-4 e-5 Solving for discharging phase by direct integration RC is constant Initial conditions (“boundary conditions”) exponentiate both sides of above right exponential decay RC = time constant = time for Q to fall to 1/e of its initial value After 3-5 time constants the action is over Time t 2t 3t 4t 5t Value e-1 e-2 e-3 e-4 e-5 % left 36.8 13.5 5.0 1.8 0.67

[] = [RC] =[(V/i)(Q/V)]=[Q/Q/t]=[t] Units for RC 8-1: We defined  = RC, which of the choices best captures the physical units for the time constant  ? [] = [RC] =[(V/i)(Q/V)]=[Q/Q/t]=[t] F (ohmfarad) C/A (coulomb per ampere) C/V (ohmcoulomb per volt) VF/A (voltfarad per ampere) s (second)

Examples: discharging capacitor C through resistor R a) When has the charge fallen to half of it’s initial value Q0? set: take log: b) When has the stored energy fallen to half of its original value? recall: and where at any time t: evaluate for: take log: c) How does the power delivered by C vary with time? power: recall: C supplies rather than absorbs power Drop minus sign power supplied by C:

RC Circuit: solution for charging Circuit Equation: Loop Equation is : Substitute : First order differential equation again: form is Q’ = - kQ + constant Same equation as for discharge, but with i0 = E / R added on right side At t = 0: Q = 0 when i = i0. Large current flows (C acts like a plain wire) As t  infinity: Current  0 (C acts like a broken wire) Q  Qinf = CE = limiting charge Solution: Charge starts from zero, grows as a saturating exponential. RC = t = TIME CONSTANT describes time dependance again Q(t)  0 as t  0 Q(t)  Qinf as t  infinity t 2t 3t Qinf Q

RC Circuit: solution for charging, continued Voltage across C while charging: Voltage across C also starts from zero and saturates exponentially Current decays exponentially just as in discharging case Growing potential Vc on C blocks current completely at t = infinity Current in the charging circuit: At t=0 C acts like a wire. At t=infinity C acts like a broken wire Voltage drop VR across the resistor: Voltage across R decays exponentially, reaches 0 as t infinity 6t 5t 4t 3t 2t t Time .998 .993 .982 .95 .865 .63 Factor Form factor: 1 – exp( - t / t ) After 3-5 time constants the action is over

RC circuit – multiple resistors at t = 0 8-2: Consider the circuit shown, The battery has no internal resistance. The capacitor has zero charge. Just after the switch is closed, what is the current through the battery? 0. /2R. 2/R. /R. impossible to determine C R R

RC circuit – multiple resistors at t = infinity 8-3: Consider the circuit shown. The battery has no internal resistance. After the switch has been closed for a very long time, what is the current through the battery? 0. /2R. 2/R. /R. impossible to determine C R R

Discharging Example: A 2 mF capacitor is charged and then connected in series with a resistance R. The original potential across it drops to ¼ of it’s starting value in 2 seconds. What is the value of the resistance? Set: Use: Define: 1 MW = 106 W Take natural log of both sides, t = 2 s:

Example: Discharging C = 500 mF R = 10 KW E = 12 V Capacitor C is charged for a long time to E, then discharged from t=0 onward. a) Find current at t = 0 b) When does VCap (voltage on C) reach 1 Volt? c) Find the current in the resistor at that time

Did not need specific values of RC Charging Example: How many time constants does it take for an initially uncharged capacitor in an RC circuit to become 99% charged? Use: Require: Take natural log of both sides: Did not need specific values of RC

E R C Example: Charging a 100 mF capacitor in series with a 10,000 W resistor, using EMF E = 5 V. How long after voltage is applied does Vcap(t) reach 4 volts? Take natural log of both sides: b) What’s the current through R at t = 2 sec?

Example: Multiple loops and EMFs Open switch S for a long time. Capacitor C charges to potential of battery 2 Then close S for a long time What is the CHANGE in charge on C? First: E2 charges C to have: Second: Close switch for a long time At equilibrium, current i3 though capacitor  zero Find outer loop current i = i1 = 12 using loop rule Now find Voltage across C, same as voltage across right hand (or left hand) branch Final charge on C: